over a finite dimensional space of polynomial maps
(see Appendix  and
[1]}, [2]}).
Three aspects of (REF ) are
particularly important.
First,
we only need to evaluate \(\pi _{\widehat{S}}\) 
up to a normalizing constant.
Second, the number \(N\)  of samples from \(\mathcal {N}({\bf 0}, {\bf I}_n)\) 
need not be limited by the cardinality of the
forecast ensemble,
in contrast to (REF ).
Third, we must evaluate the likelihood function \(\pi _{Y|X}\)  for each sample in (REF ). The latter is a crucial contrast with the stochastic map filter of Section , which only requires sampling from (i.e., simulating) the likelihood.

